Multivariate vector valued Beurling algebra analogues of theorems of Wiener and \.Zelazko
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초록
If $0<p\leq1$, $\omega$ is a weight on $\mathbb{Z}^2$, $\mathcal{A}$ is a unital complex Banach algebra and if $f$ is a continuous $\mathcal{A}$ valued function on $\mathbb{T}^2$ such that $\sum_{(m,n)\in\mathbb{Z}^2}\|\widehat{f}(m,n)\|^p\omega(m,n)<\infty$, where $\widehat{f}(m,n)$ are Fourier coefficients of $f$, and $f(z,w)$ is left invertible in $\mathcal{A}$ for all $(z,w)\in\mathbb{T}^2$, then it is shown that there is a weight $\nu$ on $\mathbb{Z}^2$ and a continuous function $g:\mathbb{T}^2\to\mathcal{A}$ such that $1\leq\nu\leq\omega$, $\nu$ is constant if and only if $\omega$ is constant, $\sum_{(m,n)\in\mathbb{Z}^2}\|\widehat{g}(m,n)\|^p\omega(m,n)<\infty$ and $g$ is a left inverse of $f$. A similar result is obtained for a continuous function $f$ from $\mathbb{T}^\infty$ (the countable copies of $\mathbb{T}$) to $\mathcal{A}$.
Abstract
If $0<p\leq1$, $\omega$ is a weight on $\mathbb{Z}^2$, $\mathcal{A}$ is a unital complex Banach algebra and if $f$ is a continuous $\mathcal{A}$ valued function on $\mathbb{T}^2$ such that $\sum_{(m,n)\in\mathbb{Z}^2}\|\widehat{f}(m,n)\|^p\omega(m,n)<\infty$, where $\widehat{f}(m,n)$ are Fourier coefficients of $f$, and $f(z,w)$ is left invertible in $\mathcal{A}$ for all $(z,w)\in\mathbb{T}^2$, then it is shown that there is a weight $\nu$ on $\mathbb{Z}^2$ and a continuous function $g:\mathbb{T}^2\to\mathcal{A}$ such that $1\leq\nu\leq\omega$, $\nu$ is constant if and only if $\omega$ is constant, $\sum_{(m,n)\in\mathbb{Z}^2}\|\widehat{g}(m,n)\|^p\omega(m,n)<\infty$ and $g$ is a left inverse of $f$. A similar result is obtained for a continuous function $f$ from $\mathbb{T}^\infty$ (the countable copies of $\mathbb{T}$) to $\mathcal{A}$.